COMPOSITIONAL THERMODYNAMICS

Giulio ChiribellaDepartment of Computer Science, The University of Hong Kong,

CIFAR-Azrieli Global Scholars ProgramCarlo Maria Scandolo, Department of Computer Science,

The University of Oxford

Compositionally Workshop, Simons Institute for the Theory of ComputingDecember 5-9 2016

THERMODYNAMICS

Thermodynamics provides a general paradigm that can be applied to different physical theories(e.g. classical mechanics, relativistic mechanics, quantum mechanics).

Is this paradigm universal? Which conditions must a physical theory satisfy in order to allow for a sensible thermodynamics?

FROM DYNAMICSTO THERMODYNAMICS

Statistical mechanics reduces thermodynamical to dynamical + probabilistic notions

• the initial state of the system is chosen at random according to a probability distribution e.g.

or

• thermodynamical quantities are expectation values

IN THE CLASSICAL WORLD:TENSION AT THE FOUNDATIONS

In the classical world of Newton and Laplace, there is no place for probability at the fundamental level.

{single-system,non-compositionalapproaches

Why should measured quantities depend on the expectations of an agent?

Attempted explanations:-ergodic theory, -symmetries-max ent principle-...

ENTERS COMPOSITIONALITY:THE QUANTUM CASE

In quantum theory, probability has a different status: mixed states can be modelled as marginals of pure states | iAB

⇢A⇢B

⇢A = TrB [ | ih |] ]

This property is called “purification”

PURIFICATION AS A FOUNDATION

Statistical ensembles frompure states of the system + its environment

Popescu-Short-Winter 2006,Gemmer-Michel-Mahler 2006,...

Idea of this work: Abstract from quantum mechanics, build an axiomatic foundation for statistical mechanics in general theories.

WHAT IS COMPOSITIONAL HERE?

2) The axioms: specify properties of composition, how physical systems are combined together

1) The framework: “general physical theories” built on symmetric monoidal categories, inspired by Abramsky and Coecke’s Categorical Quantum Mechanics.

PLAN OF THE TALK

1) The framework

2) The axioms

3) The results

THEFRAMEWORK:

OPERATIONAL-PROBABILISTIC THEORIES

Chiribella, D’Ariano, Perinotti, Probabilistic Theories with Purification, Phys. Rev. A 81, 062348 (2010)

FRAMEWORKS FOR GENERAL PHYSICAL THEORIES

Single-system (non-compositional)

• Mackey• Ludwig• Gudder • Piron• Holevo…

Composite systems

• Hardy 2001• Abramsky-Coecke 2004• D’Ariano 2006• Barrett 2006• Wilce-Barnum 2007• CDP 2009, Hardy 2009

OPERATIONAL STRUCTURE

In a nutshell:Strict symmetric monoidal category.Objects = physical systemsMorphisms = (non-deterministic) physical processes

SYSTEMS AND TESTS

-Systems: A, B, C, ... , I = trivial system (nothing)

-Tests: a test represents a (non-deterministic) physical process

A B

input-output arrow

B : output system

Ci

i: outcome, in some outcome set X : possible transformation, graphically represented as

A BCi

{Ci}i2X

A : input system

PREPARATIONS AND MEASUREMENTS

Special cases of tests: • trivial input: preparation

Bρi

Aaiai : “effect”

ρi

• trivial output: (demolition) measurement

: “state”

-Sequential composition of tests:

SEQUENTIAL COMPOSITION

=A B

{Ci}i2X

C{Dj}j2Y

C{Dj � Ci}(i,j)2X⇥Y

A

...induces sequential composition of processes:

A CiC

Dj =A CDj ◦ Ci

B

Identity test on system A = doing nothing on system A

It is a test with a single outcome

IDENTITY

A A{IA}

where is the identity process, defined by the relations

A A B

C A A

IA CiA B

Ci =

=

IA

C ADj IADj

8B, 8Ci

8C, 8Dj

-Composite systems:

PARALLEL COMPOSITION

-Parallel composition of tests:

=

A A’{Ci}i2X

B’{Dj}j2Y

B

A’A

{Ci ⌦Dj}(i,j)2X⇥Y

B B’

...induces parallel composition of processes (same diagram, without brackets)

A⌦B (A⌦ I = I ⌦A = A )

CIRCUITS

Ci1 Ci2CiN

A B C

input-output arrow

OPERATIONAL THEORY: a theory of devices that can be mounted to form circuits.

Dj1 DjN⇢j0

PROBABILISTIC STRUCTURE

PROBABILITY ASSIGNMENT

ρi ajA

=

• Preparation + measurement = probability distribution

p(aj , ρi)

p(aj , ρi) ≥ 0∑

i∈X

∑

j∈Y

p(aj , ρi) = 1{

INDEPENDENT EXPERIMENTS

• Experiments performed in parallel are statistically independent:

ρi ajA

Bσk bl

= p(aj , ρi)p(bl, σk)

e.g. the roll of two dice

OPERATIONAL-PROBABILISTIC THEORIES

OPERATIONAL-PROBABILISTIC THEORIES (OPTS)

Examples:

-classical theory-quantum theory-quantum theory on real Hilbert spaces-...

Operational-probabilistic theory =

operational structure+

probabilistic structure

QUOTIENT THEORIES

A BCi

A BDjand

are statistically equivalent iff

A BCi

R

A B

R⇢k ⇢kMl Ml=

Dj

8R, 8⇢k, 8Ml

The quotient yields a new OPT: the quotient theory

ORDERED VECTOR SPACE STRUCTURE

Theorem: In the quotient theory

• the processes of type span a real ordered vector space

• the sequential composition of two processes is linear in both arguments

A ! B

X

i

xi Ci

!�

0

@X

j

yj Dj

1

A =X

i,j

xiyj (Ci �Dj)

TransfR(A ! B)

FINITARY ASSUMPTION

In this talk:

restrict our attention to finite systems, i.e. systems for which

KEY NOTIONS

REVERSIBLE PROCESSES

A BT is reversible iff exists

A B

B AS

A A=

B A

T

B

SB=TS

such that

PURE PROCESSES

A BT is pure iff A BT A B=X

x

Tx

implies A BTx

A BT/ 8x

special case: pure states

is pure iffAX

x

=A A

implies A / A 8x

AXIOMS

AXIOM 1: CAUSALITY

Operations performed in the future cannot affect the probabilityof outcomes of experiments done in the present

e.g. in QT: ρ = Tr[ρ]

⇢A := TrA[⇢AB ]

Tr

Equivalently: there is only one way to discard a system

A Tr

AXIOM 2: PURITY PRESERVATION

Purity Preservation: the composition of pure transformations is pure

T

S pure

pure,

T

S pure

=)Special case: the product of two pure states is a pure state.

AXIOM3: PURE SHARPNESS

Pure Sharpness: for every system, there exist at least one pure effect that happens with probability 1 on some state.

A a such that

• a is pure

•

for some state

A a = 1

AXIOM 4: PURIFICATION

• Existence: For every state of A

AA=

•Uniqueness: two purifications of the same state are equivalent up to a reversible transformation

A AΨ=B B

=⇒Ψ′

BΨ′

A=

AΨ B B

U

ρ

ρ

there is a system B and a pure state of such that

B

A⌦B

Tr

TrTr

PURIFICATION: EQUIVALENT FORMULATION

A A’C =A A’

Uϕ0

E’E

Theorem: For every process there exist environments E and E’a pure state of E , and a reversible process from AE to A’E’ such that

This simulation is unique up to reversible processes on the environment.

Tr

EXAMPLES OF THEORIES SATISFYING THE AXIOMS

• quantum theory

• real vector space quantum theory

• other variants of quantum theory

classicalbit

non-classicalbit

• classical theory (!) suitably extended with non-classical systems

RESULTS

Chiribella and Scandolo,Entanglement as an axiomatic foundation for statistical mechanics arXiv:1608.04459

THE DIAGONALIZATION THEOREM

Thm: In a theory satisfying the 4 axioms, every state can be diagonalized, i.e. decomposed as

(the “spectrum”)

The spectrum is uniquely defined, the diagonalization is canonical.

Corollary: There exist distinguishable states.

THE STATE-EFFECT DUALITY

Thm: For every system A, there exists a linear, order-preserving isomorphism between and

For every pure normalized state ,one has

A = 1 (cf. Hardy’s Sharpness Axiom)

OBSERVABLES

Observables := elements of

Thm. Every observable H, can be diagonalized as

FUNCTIONAL CALCULUS

Given an observable

and a function

one can define a new observable

But what is the interpretation of this state?

Formally, we have all the tools required to define the Gibbs state:

PURITY

THE RARE PARADIGM

An agent has partial control on the parameters of a reversible dynamics:

Prepare,Discard

Ux

px

As a result, she implements a Random Reversible (RaRe) transformation

R =X

x

px

Ux

RaRe transformations = monoidal subcategory of the category of transformations

THE PURITY PREORDER

A⇢ purer than A�

iff

Definition:

A⇢RaRe A�

cf. group majorization by Masanes & Müller

Proposition. The pure states are maxima, the invariant state is the minimum.

PURITY MONOTONES AND ENTROPIES

A⇢ purer than A�

Definition: is a purity monotone if

implies

Definition: A family of functions

is an entropy if

• is a purity monotone

•

EXAMPLE

Shannon/von Neumann entropy:

A

ENTROPY/ENERGYTRADEOFF

CHARACTERIZATION OF THE GIBBS STATE

Proposition: In every theory satisfying the 4 axioms, the Gibbs state

is the state with maximum S/vN entropy among the states with given expectation value of H.

LANDAUER’S PRINCIPLE

S S’U

E’E

Suppose that the system interacts reversiblywith an environment in the Gibbs state

⇢

In every theory satisfying the 4 axioms, one has

CONCLUSIONS

4 axioms for thermodynamics

• Causality

• Purity Preservation

• Pure Sharpness

• Purification

Obtain diagonalization, entropies, Gibbs states, and Landauer’s principle.

Conjecture: every physical theory admitting a thermodynamical description can be extended to a theory that satisfies the 4 axioms.

REFERENCES FOR THIS TALK

• Chiribella, D’Ariano, Perinotti, Probabilistic Theories with Purification Phys. Rev. A 81, 062348 (2010)• Informational Derivation of Quantum Theory Phys. Rev. A 84, 012311 (2011) • Quantum from Principles, in Chiribella and Spekkens (eds), Quantum Theory: information-theoretic foundations and foils, Springer Verlag (2016)

• Chiribella and Scandolo, Entanglement and thermodynamics in general probabilistic theories New J. Phys. 17 103027 (2015) • Operational axioms for diagonalizing states EPTCS 195 96 (2015)• Entanglement as an axiomatic foundation for statistical mechanics arXiv:1608.04459• Purity in microcanonical thermodynamics: a tale of three resource theories arXiv:1608.04460